which served as generalised metric in the \(\mathbf {O}(D,D)\)  manifest formulations of both string worldsheet actions [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]} and target spacetime effective descriptions currently called Double Field Theory (DFT) [9]}, [10]}, [11]}, [12]}, [13]}, [14]}. By taking
not \(\lbrace g,B,\phi \rbrace \)  but the \(\mathbf {O}(D,D)\)  multiplets themselves as the fundamental variables, DFT opens a new avenue beyond the Riemannian paradigm [15]}, [16]}, [17]}, [18]}. In this approach, \(e^{-2d}\)  is an elementary scalar density with a unit diffeomorphic weight, and \({\cal H}_{AB}\)  satisfies its own defining properties,
\(\begin{array}{ll}{\cal H}_{AB}={\cal H}_{BA}\,,\qquad &\qquad {\cal H}_{A}{}^{C}{\cal H}_{B}{}^{D}{\cal J}_{CD}={\cal J}_{AB}\,.\end{array}\) 
